<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>We consider the approximate solution of the control problem with minimum energy for an object described by the heat equation, with the process described by the linear equation of parabolic type and the system controlled by impulsive external influences. Our optimal control problem deals with finding a control parameter belonging to the class of admissible controls that provides the desired temperature distribution in a finite time with minimal energy consumption (energy consumption is described by the quadratic functional). Previous works dedicated to optimal impulse control problems have mostly used the Pontryagin’s maximum principle. However, from a practical point of view, this approach does not lead to satisfactory results. This is due to the fact that the corresponding boundary value problems in this case have no solution in a traditional class of absolutely continuous trajectories. In this work, we propose a method based on the moment relations. We seek for the approximate solution of the corresponding boundary value problem in the form of finite Fourier sum and state our optimal control problem in a finite-dimensional phase space. As a result, we obtain an optimal impulse control problem in a finite-dimensional function space. Taking into account the given condition for a finite time, we reduce the obtained problem to the L-problem of moments. Thus, the problem of finding a control parameter is reduced to the solution of the system of Fredholm integral equations of the first kind, with the norm of the sought solution not exceeding a given number. By Levi’s theorem, every element of Hilbert space can be represented by the sum of the elements of two orthogonal subspaces. This assertion makes it possible to find control parameters in analytical form. We also establish the convergence of the chosen approximation.</p>
</abstract>

We consider the approximate solution of the control problem with minimum energy for an object described by the heat equation, with the process described by the linear equation of parabolic type and the system controlled by impulsive external influences. Our optimal control problem deals with finding a control parameter belonging to the class of admissible controls that provides the desired temperature distribution in a finite time with minimal energy consumption (energy consumption is described by the quadratic functional). Previous works dedicated to optimal impulse control problems have mostly used the Pontryagin’s maximum principle. However, from a practical point of view, this approach does not lead to satisfactory results. This is due to the fact that the corresponding boundary value problems in this case have no solution in a traditional class of absolutely continuous trajectories. In this work, we propose a method based on the moment relations. We seek for the approximate solution of the corresponding boundary value problem in the form of finite Fourier sum and state our optimal control problem in a finite-dimensional phase space. As a result, we obtain an optimal impulse control problem in a finite-dimensional function space. Taking into account the given condition for a finite time, we reduce the obtained problem to the L-problem of moments. Thus, the problem of finding a control parameter is reduced to the solution of the system of Fredholm integral equations of the first kind, with the norm of the sought solution not exceeding a given number. By Levi’s theorem, every element of Hilbert space can be represented by the sum of the elements of two orthogonal subspaces. This assertion makes it possible to find control parameters in analytical form. We also establish the convergence of the chosen approximation.

Approximate Solution of Optimal Pulse Control Problem Associated with the Heat Conduction Process

Azerbaijan State Oil and Industry University

Proceedings of the Latvian Academy of Sciences. Section B. Natural, Exact, and Applied Sciences.

This work is licensed under the Creative Commons Attribution 4.0 International License.

We consider the approximate solution of the control problem with minimum energy for an object described by the heat equation, with the process described by the...

Approximate Solution of Optimal Pulse Control Problem Associated with the Heat Conduction Process

Published Online: Jan 06, 2024

Page range: 263 - 270

Received: Nov 24, 2023

Accepted: Nov 24, 2023

DOI: https://doi.org/10.2478/prolas-2023-0030

Keywords
optimal impulse control of parabolic system, heat transfer process, Fourier method, moment problem

© 2023 Rashad Mammadov et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Approximate Solution of Optimal Pulse Control Problem Associated with the Heat Conduction Process

Published Online: Jan 06, 2024

Page range: 263 - 270

Received: Nov 24, 2023

Accepted: Nov 24, 2023

DOI: https://doi.org/10.2478/prolas-2023-0030

Keywordsoptimal impulse control of parabolic system, heat transfer process, Fourier method, moment problem

© 2023 Rashad Mammadov et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
optimal impulse control of parabolic system, heat transfer process, Fourier method, moment problem